Electronic instruments, such as wide-band electronic instruments used in the test and measurement of electronic devices under test (DUTS), typically need to be calibrated. Examples of electronic instruments are 1) an arbitrary waveform generator (ARB), the core of which typically comprises a digital-to-analog converter (DAC), or 2) a receiver such as an oscilloscope, the core of which typically comprises an analog-to-digital converter (ADC). Around the core functionality of an electronic instrument, there typically exist a number of other devices, such as conditioning electronics (e.g., filters and amplifiers) and frequency translation devices (e.g., mixers for up-conversion and down-conversion of signals of interest).
A signal that is transmitted from, to or between electronic instruments can be processed through several stages in its transmission or reception. For instance, in a typical application, such as the testing of a radio frequency (RF) amplifier, an initial test signal could be generated by a baseband ARB, filtered, upconverted by a mixer, and then subjected to further amplification and filtering before its application to a DUT. An output test signal that is generated by the DUT, in response to the initial test signal, may then be received by a measurement instrument such as an oscilloscope or spectrum analyzer. Due to non-idealities in the electronics of the transmission and reception paths, the signals transmitted to and from the DUT may undergo distortions. Calibration refers to the processes that are applied to physical or mathematical signal representations in an attempt to remove or minimize these distortions.
Distortion of a signal may be classified as linear or nonlinear. Linear calibration of an electronic instrument is common, and is typically accomplished by adjusting the phase and amplitude of an excitation signal. This compensation is perhaps best understood in the frequency domain. That is, any signal can be represented in the frequency domain by considering a Fourier deposition (either a continuous decomposition using the Fourier Transform, or a discrete decomposition using the so-called discrete Fourier transform (DFT) of the signal of interest. See, e.g., Julius O. Smith III, “Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications—Second Edition”, W3K Publishing (2007). By a fundamental property of linear systems, linear distortion results in a signal with exactly the same frequency components as the undistorted signal—however, linear distortion can cause a shift in the phase and amplitude of each frequency component. When an input signal is represented (in the frequency domain) as a collection of complex numbers known as “phasors,” linear distortion can be viewed as a transformation that scales and rotates each input phasor. In the time domain, linear calibration is typically accomplished by the construction of a finite impulse response (FIR) filter, which is constructed to satisfy frequency domain constraints. In practice, an ‘impulse response’ is typically used to estimate the linear distortion of a signal, and an appropriate FIR filter is used for linear calibration. Because of the importance of linear calibration procedures, the National Institute of Standards (NIST), has studied the linear calibration problem in detail and set forth standards for linear calibration processes and signals for some measurement instruments. See, e.g., William L. Gans, “Dynamic Calibration of Waveform Recorders and Oscilloscopes Using Pulse Standards”, IEEE Transactions on Instrumentation and Measurement, Vol. 39, No. 6, pp. 952-957 (December 1990).
A hallmark of nonlinear distortion is the creation of energy at frequencies distinct from the frequency of the input signal. In one form, nonlinear distortion can result from a so-called squaring law. That is, if a signal has energy at a frequency f1, then the squaring process in the time domain y(t)=x(t)2 leads to energy, or signal distortion, at twice the input frequency (i.e., at 2*f1=f2). Informally, nonlinear distortion leads to so-called “spurs” in the frequency domain. These spurs are easily viewed using a power spectrum analyzer. See, e.g., “Agilent PSA Performance Spectrum Analyzer Series—Optimizing Dynamic Range for Distortion Measurements”, Agilent Technologies, Inc., (2000).
The importance of spurs in limiting the performance of electronic instruments is well-known, and is commonly quantified by the so called “spurious free dynamic range” (SFDR), usually provided as a key metric for measurement instrument performance. SFDR is also the subject of various Institute of Electrical and Electronics Engineers (IEEE) standards for uniform measurement. See, e.g., E. Balestrieri, et al., “Some Critical Notes on DAC Frequency Domain Specifications”, XVIII Imeko World Congress (2006). Consideration of higher order nonlinear interactions (e.g., third order interactions) shows that nonlinear distortion can also cause distortion at or near the input excitation frequencies. This distortion is in addition to any linear distortion, and is often referred to as “intermodulation distortion”, because it arises from signal mixing processes inherent in nonlinear signal interactions. Intermodulation distortion can complicate signal transmission and reception considerably, because it needs to be untangled from the underlying signal and its linear distortion.
Real-world electronic systems are also subject to a range of effects (e.g. oscillator feed through, electronic component operation, dependencies on temperature) that can cause a wide range of signal impairments and distortions that often need to be systematically accounted for and corrected (or avoided) during a calibration process. A brief overview of some of these effects and considerations for ARBs is described by Mike Griffin, et al. in “Conditioning and Correction of Arbitrary Waveforms—Part 1: Distortion”, High Frequency Electronics, pp. 18-28 (August 2005) and in “Conditioning and Correction of Arbitrary Waveforms—Part 2: Other Impairments”, High Frequency Electronics, pp. 18-26 (September 2005).